Basically they're claiming that a very distant (> 500 AU) brown dwarf or companion star can pump up the eccentricity of a planet close to the primary star, if the companion is on a very inclined (> 40°) orbit. This sounds crazy to me - but what's crazier is that the magnitude of the eccentricity pumping is determined solely by the difference in inclination of the orbits, and not by mass or distance or anything (those just determine the period of the oscillation).

I tried running this overnight with a 65k timestep, and recorded the first million year's worth of evolution, and I got no noticeable change in eccentricity of the inner gas giant (beyond the usual small scale cycles), and a very very slight increase in its inclination. I just got some more data points for around the 10 million year mark and the inclination of the gas giant has gone from 0.00 (at the start) to 0.02 (at 1 million years) to 0.24 degrees (at 11 million years) - the eccentricity hasn't changed though.

I've heard of this before. It's related to the Kozai mechanism. I have not yet simulated it for binary stars, but on the "Simulations" link there is an article called "Kozai Mechanism". It shows 2 examples with Gravity Simulator. The first places the Moon in a circular polar orbit around Earth. All seems fine for a few orbits, but then eccentricity is pumped into the Moon's orbit. Within a decade, its perigee is below the surface of the Earth

The next example doesn't actually simulate the effect, but shows what is theorized to be caused by the effect. The jovian system of moons contains no moons with inclinations above about 60 degrees. The reasoning... If they existed, the Kozai mechanism would pump eccentricity into them until their perijoves were in the vicinity of the Galilean Moons. Then their days are seriously numbered (think 5-planet system ).

Comparing a binary star system to the Earth / Moon simulation, the two stars are analgous to the Sun / Earth, and the planet is analgous to the Moon. There's no reason Gravity Simulator shouldn't be able to handle this, although the time scale required to see results may be very long.

Notice in this simulation that it takes a while for the action to begin. The Moon orbits the Earth normally for several orbits before you notice any significant eccentricity develop. But once it does, the rate accelerates.

Sometimes, when a simulation is difficult because of the amount of time needed, or the amount of objects needed, you can play some games to get good results.

For example, in the case of the Kozai Mechanism, you already know that Gravity Simulator can quickly model an exaggerated version as demonstrated by placing the Moon in a polar Earth orbit. It takes about 10 years for the eccentricity to reach 0.9. Enter this data into excel (1 AU, 10 years). Next, create a new simulation with an identical Earth / Moon system at 2 AU. Give it a go and plot your data. With only a few data points you should be able to determine if your function is growing linearly, or as a polynomial or exponential function. Graph it in Excel, add a trendline and have it display the formula for the trend line. Now you can simply extrapolate the curve / line to values too large to simulate.

Try it again, but this time instead of plotting semi-major axis against time to exceed 0.9, try initial inclination vs. time, or vs. max eccentricity. Use the same technique in Excel. Soon you'll have a series of curves that you may be able to use to extrapolate any condition.

Right, but the problem I have with this is the idea that an inlcined brown dwarf 500 AU from the star can screw around with the eccentricities of a planet orbiting really close to the star. That sounds really counter-intuitive to me.

It makes it annoying too, when you're trying to write a worldbuilding program!

That said, it only holds if the BD or companion star is on a very inclined orbit (45° or more) relative to the planet/ecliptic. I'm not entirely convinced that's likely - wouldn't they have formed in the same plane because the companions form from the same disk of material as the primary and planets?

Yes, but wide systems are not very well protected against disturbances beyond the system. Over time lots of crazy stuff can happen to distant objects. Consider Xena (aka Iris, 2003 UB313). It's inclination is almost 45 degrees.

In the system you describe, at 500 AU, its very likely that a close stellar passage when the star system was young and still in its birth cluster, would pump lots of eccentricity and inclination into the wide system.

Interesting to note that Uranus, with its almost polar tilt, doesn't display signs of the Kozai mechanism in its system of moons. Or does it?

Concerning the last question : I think the Kozai comes only in action if the inclination is big related to the plane of motion of the mother planet . The oriontation of spin around its axe doesn't matter ....

Sorry, I wasn't too clear with the way I phrased it. You're right. The tilt has nothing to do with it. But since Uranus' moons orbit in the same plane of its equator, the tilt implies that the moons' are in steep orbits with respect to the plane of Uranus' orbit.

25 million years into it, and the inclination of the gg has gone from 0.26 to 0.56 in the 14 million years or so since this morning. Eccentricity is still cycling in exactly the same way as it did before though, at 0.6 +/- 0.05 - that hasn't changed at all each time I've looked at it.

So... over time, inclination is increasing from zero, but eccentricity is just staying stable (on average).

Sorry, I wasn't too clear with the way I phrased it. You're right. The tilt has nothing to do with it. But since Uranus' moons orbit in the same plane of its equator, the tilt implies that the moons' are in steep orbits with respect to the plane of Uranus' orbit.

you're right , I overlooked this fact . Maybe the Kozai is not "visible" in this case while the orbital period of Uranus is much longer than earths . I estimate that the Kozai force must be proportional to 1/omega^2 .

Check out how sensitive the Kozai Mechanism is to distance. I ran a simulation similar to the one on the website where the Moon was placed in polar orbit around Earth. But in this sim, I had 5 Earth systems:

Earth 1: 1 AU Moon : 384,000 km Inc: 90 deg Ecc: 0

Earth 2: 1.5 AU Moon : 384,000 km Inc: 90 deg Ecc: 0

Earth 3: 2 AU Moon : 384,000 km Inc: 90 deg Ecc: 0

Earth 4: 2.5 AU Moon : 384,000 km Inc: 90 deg Ecc: 0

Earth 5: 3 AU Moon : 384,000 km Inc: 90 deg Ecc: 0

And the data:

The Moons of Earth 2-5 are all crammed together on the bottom, barely visible with the scale high enough to show Earth 1's Moon full range. I did not give the Earth's any size in this simulation to avoid collisions. So it is interesting to note that the Kozai Mechanism does not pump ecc>1 into a system. Once it approaches 1, it backs off, and I'm guessing periodically goes back to close to 0 and repeats.

Leaving the data for Earth 1's moon off the chart to do a better job of plotting the moons for Earth 2-5 gives this:

Again, the Moon from the closest Earth significantly outpaces the others in its eccentricity gain.

I think I'll run a similar sim tommorow, but with Earth SMA values more closely surrounding 1 AU.

This was run at a slow time step of 16 (not 16K !), for only about 7 years.